Tamil Nadu Board of Secondary EducationHSC Science Class 12

# Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 3 - Theory of Equations [Latest edition]

## Solutions for Chapter 3: Theory of Equations

Below listed, you can find solutions for Chapter 3 of Tamil Nadu Board of Secondary Education Tamil Nadu Board Samacheer Kalvi for Class 12th Mathematics Volume 1 and 2 Answers Guide.

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7
Exercise 3.1 [Pages 106 - 107]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.1 [Pages 106 - 107]

Exercise 3.1 | Q 1 | Page 106

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid

Exercise 3.1 | Q 2. (i) | Page 106

Construct a cubic equation with roots 1, 2 and 3

Exercise 3.1 | Q 2. (ii) | Page 106

Construct a cubic equation with roots 1, 1, and – 2

Exercise 3.1 | Q 2. (iii) | Page 106

Construct a cubic equation with roots 2, 1/2, and 1

Exercise 3.1 | Q 3. (i) | Page 106

If α, β and γ are the roots of the cubic equation x3 + 2x2 + 3x + 4 = 0, form a cubic equation whose roots are 2α, 2β, 2γ

Exercise 3.1 | Q 3. (ii) | Page 106

If α, β and γ are the roots of the cubic equation x3 + 2x2 + 3x + 4 = 0, form a cubic equation whose roots are 1/alpha, 1/beta, 1/γ

Exercise 3.1 | Q 3. (iii) | Page 106

If α, β and γ are the roots of the cubic equation x3 + 2x2 + 3x + 4 = 0, form a cubic equation whose roots are - alpha, -beta, -γ

Exercise 3.1 | Q 4 | Page 106

Solve the equation 3x3 – 16x2 + 23x – 6 = 0 if the product of two roots is 1

Exercise 3.1 | Q 5 | Page 106

Find the sum of squares of roots of the equation 2x^4 - 8x^3 + 6x^2 - 3 = 0

Exercise 3.1 | Q 6 | Page 107

Solve the equation x3 – 9x2 + 14x + 24 = 0 if it is given that two of its roots are in the ratio 3 : 2

Exercise 3.1 | Q 7 | Page 107

If α, β, and γ are the roots of the polynomial equation ax3 + bx2 + cx + d = 0, find the value of sum  alpha/(betaγ) in terms of the coefficients

Exercise 3.1 | Q 8 | Page 107

If α, β, γ and δ are the roots of the polynomial equation 2x4 + 5x3 – 7x2 + 8 = 0, find a quadratic equation with integer coefficients whose roots are α + β + γ + δ and αβγδ

Exercise 3.1 | Q 9 | Page 107

If p and q are the roots of the equation lx2 + nx + n = 0, show that sqrt("p"/"q") + sqrt("q"/"p") + sqrt("n"/l) = 0

Exercise 3.1 | Q 10 | Page 107

If the equations x2 + px + q = 0 and x2 + p’x + q’ = 0 have a common root, show that it must be equal to ("pq'" - "p'q")/("q" - "q") or ("q" - "q'")/("p'" - "P")

Exercise 3.1 | Q 11 | Page 107

A 12 metre tall tree was broken into two parts. It was found that the height of the part which was left standing was the cube root of the length of the part that was cut away. Formulate this into a mathematical problem to find the height of the part which was left standing

Exercise 3.2 [Page 112]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.2 [Page 112]

Exercise 3.2 | Q 1 | Page 112

If k is real, discuss the nature of the roots of the polynomial equation 2x2 + kx + k = 0, in terms of k

Exercise 3.2 | Q 2 | Page 112

Find a polynomial equation of minimum degree with rational coefficients, having 2 + sqrt(3)"i" as a root

Exercise 3.2 | Q 3 | Page 112

Find a polynomial equation of minimum degree with rational coefficients, having 2i + 3 as a root

Exercise 3.2 | Q 4 | Page 112

Find a polynomial equation of minimum degree with rational coefficients, having sqrt(5) - sqrt(3) as a root

Exercise 3.2 | Q 5 | Page 112

Prove that a straight line and parabola cannot intersect at more than two points

Exercise 3.3 [Page 117]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.3 [Page 117]

Exercise 3.3 | Q 1 | Page 117

Solve the cubic equation: 2x3 – x2 – 18x + 9 = 0 if sum of two of its roots vanishes

Exercise 3.3 | Q 2 | Page 117

Solve the equation 9x3 – 36x2 + 44x – 16 = 0 if the roots form an arithmetic progression

Exercise 3.3 | Q 3 | Page 117

Solve the equation 3x3 – 26x2 + 52x – 24 = 0 if its roots form a geometric progression

Exercise 3.3 | Q 4 | Page 117

Determine k and solve the equation 2x3 – 6x2 + 3x + k = 0 if one of its roots is twice the sum of the other two roots

Exercise 3.3 | Q 5 | Page 117

Find all zeros of the polynomial x6 – 3x5 – 5x4 + 22x3 – 39x2 – 39x + 135, if it is known that 1 + 2i and sqrt(3) are two of its zeros

Exercise 3.3 | Q 6. (i) | Page 117

Solve the cubic equations:

2x3 – 9x2 + 10x = 3

Exercise 3.3 | Q 6. (ii) | Page 117

Solve the cubic equations:

8x3 – 2x2 – 7x + 3 = 0

Exercise 3.3 | Q 7 | Page 117

Solve the equation:

x4 – 14x2 + 45 = 0

Exercise 3.4 [Page 118]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.4 [Page 118]

Exercise 3.4 | Q 1. (i) | Page 118

Solve: (x – 5)(x – 7) (x + 6)(x + 4) = 504

Exercise 3.4 | Q 1. (ii) | Page 118

Solve: (x – 4)(x – 2)(x- 7)(x + 1) = 16

Exercise 3.4 | Q 2 | Page 118

Solve: (2x – 1)(x + 3)(x – 2)(2x + 3) + 20 = 0

Exercise 3.5 [Page 124]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.5 [Page 124]

Exercise 3.5 | Q 1. (i) | Page 124

Solve the following equations
sin² x – 5 sin x + 4 = 0

Exercise 3.5 | Q 1. (ii) | Page 124

Solve the following equations
12x3 + 8x = 29x2 – 4 = 0

Exercise 3.5 | Q 2. (i) | Page 124

Examine for the rational roots of

2x3 – x2 – 1 = 0

Exercise 3.5 | Q 2. (ii) | Page 124

Examine for the rational roots of

x8 – 3x + 1 = 0

Exercise 3.5 | Q 3 | Page 124

Solve: 8x^(3/(2"n")) - 8x^((-3)/(2"n")) = 63

Exercise 3.5 | Q 4 | Page 124

Solve: 2sqrt(x/"a") + 3sqrt("a"/x) = "b"/"a" + (6"a")/"b"

Exercise 3.5 | Q 5. (i) | Page 124

Solve the equation
6x4 – 35x3 + 62x2 – 35x + 6 = 0

Exercise 3.5 | Q 5. (ii) | Page 124

Solve the equation
x4 + 3x3 – 3x – 1 = 0

Exercise 3.5 | Q 6 | Page 124

Find all real numbers satisfying 4x – 3(2x+2) + 25 = 0

Exercise 3.5 | Q 7 | Page 124

Solve the equation 6x4 – 5x3 – 38x2 – 5x + 6 = 0 if it is known that 1/3 is a solution

Exercise 3.6 [Page 127]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.6 [Page 127]

Exercise 3.6 | Q 1 | Page 127

Discuss the maximum possible number of positive and negative roots of the polynomial equation 9x9 – 4x8 + 4x7 – 3x6 + 2x5 + x3 + 7x2 + 7x + 2 = 0

Exercise 3.6 | Q 2 | Page 127

Discuss the maximum possible number of positive and negative roots of the polynomial equations x2 – 5x + 6 and x2 – 5x + 16. Also, draw a rough sketch of the graphs

Exercise 3.6 | Q 3 | Page 127

Show that the equation x9 – 5x5 + 4x4 + 2x2 + 1 = 0 has atleast 6 imaginary solutions

Exercise 3.6 | Q 4 | Page 127

Determine the number of positive and negative roots of the equation x9 – 5x8 – 14x7 = 0

Exercise 3.6 | Q 5 | Page 127

Find the exact number of real zeros and imaginary of the polynomial x9 + 9x7 + 7x5 + 5x3 + 3x

Exercise 3.7 [Pages 127 - 128]

### Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 3 Theory of Equations Exercise 3.7 [Pages 127 - 128]

#### MCQ

Exercise 3.7 | Q 1 | Page 127

Choose the correct alternative:
A zero of x3 + 64 is

• 0

• 4

• 4i

• – 4

Exercise 3.7 | Q 2 | Page 127

Choose the correct alternative:
If f and g are polynomials of degrees m and n respectively, and if h(x) = (f o g)(x), then the degree of h is

• mn

• m + n

• mn

• nm

Exercise 3.7 | Q 3 | Page 127

Choose the correct alternative:
A polynomial equation in x of degree n always has

• n distinct roots

• n real roots

• n complex roots

• at most one root

Exercise 3.7 | Q 4 | Page 127

Choose the correct alternative:
If α, β and γ are the zeros of x3 + px2 + qx + r, then sum 1/alpha is

• - "q"/"r"

• - "p"/"r"

• "q"/"r"

• -"q"/"p"

Exercise 3.7 | Q 5 | Page 127

Choose the correct alternative:
According to the rational root theorem, which number is not possible rational root of 4x7 + 2x7 – 10x3 – 5?

• – 1

• 5/4

• 4/5

• 5

Exercise 3.7 | Q 6 | Page 128

Choose the correct alternative:
The polynomial x3 – kx2 + 9x has three real roots if and only if, k satisfies

• |k| ≤ 6

• k = 0

• |k| > 6

• |k| ≥ 6

Exercise 3.7 | Q 7 | Page 128

Choose the correct alternative:
The number of real numbers in [0, 2π] satisfying sin4x – 2 sin2x + 1 is

• 2

• 4

• 1

• oo

Exercise 3.7 | Q 8 | Page 128

Choose the correct alternative:
If x3 + 12x2 + 10ax + 1999 definitely has a positive zero, if and only if

• a ≥ 0

• a > 0

• a < 0

• a ≤ 0

Exercise 3.7 | Q 9 | Page 128

Choose the correct alternative:
The polynomial x3 + 2x + 3 has

• one negative and two imaginary zeros

• one positive and two imaginary zeros

• three real zeros

• no zeros

Exercise 3.7 | Q 10 | Page 128

Choose the correct alternative:
The number of positive roots of the polynomials sum_("j" = 0)^"n"  ""^"n""C"_"r" (- 1)^"r" x^"r" is

• 0

• n

• < n

• r

## Solutions for Chapter 3: Theory of Equations

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7

## Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 3 - Theory of Equations

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Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 3 Theory of Equations are Introduction to Theory of Equations, Basics of Polynomial Equations, Vieta’s Formulae and Formation of Polynomial Equations, Nature of Roots and Nature of Coefficients of Polynomial Equations, Roots of Higher Degree Polynomial Equations, Polynomial Equations with No Additional Information, Polynomials with Additional Information, Descartes Rule.

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